Semi-definite form

A semi-definite form is a quadratic form over an ordered field , representing either only non-negative or only non-positive elements of the field. In the first case, the quadratic form is called non-negative , in the second - non - positive quadratic form.

Most often, semi-definite forms are considered over the field of real numbers .

Over the field of complex numbers , semidefinite (non-negative and non-positive) Hermitian quadratic forms are defined similarly.

If a $b$ ${\ displaystyle b}$ $b$ - symmetric bilinear or Hermitian form , and $q(x)=b(x,x)$ ${\ displaystyle q (x) = b (x, x)}$ ${\ displaystyle q (x) = b (x, x)}$ is a semi-definite form, then a form $b$ ${\ displaystyle b}$ $b$ also sometimes called semi-definite (non-negative or non-positive).

Properties

If a $q$ ${\ displaystyle q}$ $q$ - quadratic or Hermitian semidefinite form in vector space $V$ ${\ displaystyle V}$ $V$ then

N=\{x\in V|q(x)=0\}

{\ displaystyle N = \ {x \ in V | q (x) = 0 \}}

N=\{x\in V|q(x)=0\}

is a subspace matching the core of the form $q$ ${\ displaystyle q}$ $q$ , and on $V/N$ ${\ displaystyle V / N}$ $V/N$ a positive definite or negative definite form is naturally induced.

Semi-definite form

Properties

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